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Doubling inequalities for the Lamé system with rough coefficients

In this paper we study the local behavior of a solution to the Lamé system when the Lamé coefficients $λ$ and $μ$ satisfy that $μ$ is Lipschitz and $λ$ is essentially bounded in dimension $n\ge 2$. One of the main results is the \emph{local} doubling inequality for the solution of the Lamé system. This is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the \emph{global} doubling inequality, which is useful in some inverse problems.